# Chapter Outline. Diffusion - how do atoms move through solids?

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1 Chapter Outline iffusion - how do atoms move through solids? iffusion mechanisms Vacancy diffusion Interstitial diffusion Impurities The mathematics of diffusion Steady-state diffusion (Fick s first law) Nonsteady-State iffusion (Fick s second law) Factors that influence diffusion iffusing species Host solid Temperature Microstructure

2 What is diffusion? iffusion is material transport by atomic motion. results of an atomistic simulation of atomic mixing homework project in MSE 4592/6270 Inhomogeneous materials can become homogeneous by diffusion. For an active diffusion to occur, the temperature should be high enough to overcome energy barriers to atomic motion. 2

3 Interdiffusion and Self-diffusion Interdiffusion (or impurity diffusion) occurs in response to a concentration gradient. (Heat) Before After Self-diffusion is diffusion in one-component material, when all atoms that exchange positions are of the same type. 3

4 iffusion Mechanisms (I) Vacancy diffusion mechanism Atom migration Vacancy migration Before After To jump from lattice site to lattice site, atoms need energy to break bonds with neighbors, and to cause the necessary lattice distortions during jump. This energy comes from the thermal energy of atomic vibrations (E av ~ k B T) The direction of flow of atoms is opposite the vacancy flow direction. 4

5 iffusion Mechanisms (II) Interstitial diffusion mechanism Interstitial atom before jump Interstitial atom after jump Interstitial diffusion is generally faster than vacancy diffusion because bonding of interstitials to the surrounding atoms is normally weaker and there are many more interstitial sites than vacancy sites to jump to. Requires small impurity atoms (e.g. C, H, O) to fit into interstices in host. 5

6 iffusion Flux The flux of diffusing atoms, J, is used to quantify how fast diffusion occurs. The flux is defined as either the number of atoms diffusing through unit area per unit time (atoms/m 2 -second) or the mass of atoms diffusing through unit area per unit time, (kg/m 2 - second). For example, for the mass flux we can write J M / At (/A) (dm/dt) (Kg m -2 s - ) where M is the mass of atoms diffusing through the area A during time t. A J 6

7 Steady-State iffusion Steady state diffusion: the diffusion flux does not change with time. Concentration profile: concentration of atoms/molecules of interest as function of position in the sample. Concentration gradient: dc/dx (Kg m -4 ): the slope at a particular point on concentration profile. dc dx ΔC Δx C x A A C x B B 7

8 Fick s first law: the diffusion flux along direction x is proportional to the concentration gradient J Steady-State iffusion: Fick s first law dc dx where is the diffusion coefficient The concentration gradient is often called the driving force in diffusion (but it is not a force in the mechanistic sense). The minus sign in the equation means that diffusion is down the concentration gradient. 8

9 iffusion down the concentration gradient J dc dx Why do the random jumps of atoms result in a flux of atoms from regions of high concentration towards the regions of low concentration? Atoms here jump randomly both right and left But there are not many atoms here to jump to the left As a result there is a net flux of atoms from left to right. 9

10 Nonsteady-State iffusion: Fick s second law In many real situations the concentration profile and the concentration gradient are changing with time. The changes of the concentration profile can be described in this case by a differential equation, Fick s second law. C t x C x 2 C 2 x Solution of this equation is concentration profile as function of time, C(x,t): 0

11 Nonsteady-State iffusion: Fick s second law C t 2 C 2 x Fick s second law relates the rate of change of composition with time to the curvature of the concentration profile: C C C x x x Concentration increases with time in those parts of the system where concentration profile has a positive curvature. And decreases where curvature is negative. The phenomenological description based on the Fick s laws is valid for any atomic mechanism of diffusion. Understanding of the atomic mechanisms is important, however, for predicting the dependence of the atomic mobility (and, therefore, diffusion coefficient) on the type of interatomic bonding, temperature, and microstructure.

12 iffusion Thermally Activated Process (I) (not tested) To jump from lattice site to lattice site, atoms need energy to break bonds with neighbors, and to cause the necessary lattice distortions during jump. The energy necessary for motion, E m, is called the activation energy for vacancy motion. Energy Atom E m Vacancy istance Schematic representation of the diffusion of an atom from its original position into a vacant lattice site. Activation energy E m has to be supplied to the atom so that it could break inter-atomic bonds and to move into the new position. 2

13 iffusion Thermally Activated Process (II) (not tested) The average thermal energy of an atom (k B T ev for room temperature) is usually much smaller that the activation energy E m (~ ev/atom) and a large fluctuation in energy (when the energy is pooled together in a small volume) is needed for a jump. The probability of such fluctuation or frequency of jumps, R j, depends exponentially on temperature and can be described by equation that is attributed to Swedish chemist Arrhenius: R j R exp E m 0 k BT where R 0 is so-called attempt frequency proportional to the frequency of atomic vibrations. 3

14 iffusion Thermally Activated Process (III) (not tested) For the vacancy diffusion mechanism the probability for any atom in a solid to move is the product of the probability of finding a vacancy in an adjacent lattice site (see Chapter 4): Q P z exp v kbt R j R exp E m 0 k BT The diffusion coefficient can be estimated as z is coordination number (number of atoms adjacent to the vacancy) and the frequency of jumps (probability of thermal fluctuation needed to overcome the energy barrier for vacancy motion): zr 0 0 a 2 E exp m k B T ( E Q ) + exp m V Q exp k B T V k 0 B T exp Q d k B T Temperature dependence of the diffusion coefficient, follows the Arrhenius dependence. 4

15 iffusion Temperature ependence (I) J dc dx iffusion coefficient is the measure of mobility of diffusing species. Q exp d 0 RT 0 temperature-independent preexponential (m 2 /s) Q d the activation energy for diffusion (J/mol or ev/atom) R the gas constant (8.3 J/mol-K or ev/atom-k) T absolute temperature (K) The above equation can be rewritten as ln Qd ln0 R T log Qd or log0 2.3R T The activation energy Q d and preexponential 0, therefore, can be estimated by plotting ln versus /T or log versus /T. Such plots are called Arrhenius plots. 5

16 iffusion Temperature ependence (II) b log 0 y ax+ b a Qd 2.3R x /T Graph of log vs. /T has slop of Q d /2.3R, intercept of log o log log 0 Qd 2.3R T Q d log 2.3R T log T 2 2 6

17 Temperature dependence: Interstitial and vacancy diffusion mechanisms (I) log log0 Q d 2.3R T iffusion of interstitials is typically faster as compared to the vacancy diffusion mechanism (self-diffusion or diffusion of substitutional atoms). 7

18 Temperature dependence: Interstitial and vacancy diffusion mechanisms Q d 0exp k BT Impurity 0, mm 2 /s - Q d, kj/mol Interstitial diffusion mechanism C in FCC Fe C in BCC Fe N in FCC Fe N in BCC Fe H in FCC Fe H in BCC Fe Smaller atoms cause less distortion of the lattice during migration and diffuse more readily than big ones (the atomic diameters decrease from C to N to H). iffusion is faster in open lattices or in open directions Vacancy diffusion mechanism Fe in FCC Fe Fe in BCC Fe Ni in Cu Si in Si 0, mm 2 /s Q d, kj/mol

19 Example: Temperature dependence of At 300ºC the diffusion coefficient and activation energy for Cu in Si are (300ºC) 7.8 x 0 - m 2 /s Q d 4.5 kj/mol What is the diffusion coefficient at 350ºC? transform data ln Temp T /T Q d 0 ln ln0 R T Q d exp k BT ln 2 ln 0 Q R d T2 and ln ln 0 Q R d T ln 2 ln ln 2 Q R d T2 T 9

20 Example: Temperature dependence of (continued) ln Q R T2 T 2 d 2 exp Q R d T2 T T K T K 2 (7.8 x 0 m 2 /s) 4,500 J/mol exp 8.34 J/mol - K 623 K 573 K x 0 - m 2 /s 20

21 iffusion: Role of the microstructure (I) Self-diffusion coefficients for Ag depend on the diffusion path. In general the diffusivity if greater through less restrictive structural regions grain boundaries, dislocation cores, external surfaces. 2

22 iffusion: Role of the microstructure (II) The plots below are from the computer simulation by T. Kwok, P. S. Ho, and S. Yip. Initial atomic positions are shown by the circles, trajectories of atoms are shown by lines. We can see the difference between atomic mobility in the bulk crystal and in the grain boundary region. 22

23 Example: iffusion in nanocrystalline materials Lin et al. J. Phys. Chem. C 4, 5686, 200 Arrhenius plots for 59 Fe diffusivities in nanocrystalline Fe and other alloys compared to the crystalline Fe (ferrite). [Wurschum et al. Adv. Eng. Mat. 5, 365, 2003] 23

24 Factors that influence diffusion: Summary Temperature - diffusion rate increases very rapidly with increasing temperature iffusion mechanism diffusion by interstitial mechanism is usually faster than by vacancy mechanism iffusing and host species - o, Q d are different for every solute, solvent pair Microstructure - diffusion is faster in polycrystalline materials compared to single crystals because of the accelerated diffusion along grain boundaries. 24

25 iffusion in material processing Case Hardening: Hardening the surface of a metal by exposing it to impurities that diffuse into the surface region and increase surface hardness. Common example of case hardening is carburization of steel. iffusion of carbon atoms (interstitial mechanism) increases concentration of C atoms and makes iron (steel) harder. oping silicon with phosphorus for n-type semiconductors. Process of doping:. eposit P rich layers on surface. silicon 2. Heat it. 3. Result: oped semiconductor regions. silicon 25

26 Summary Make sure you understand language and concepts: Activation energy Concentration gradient iffusion iffusion coefficient iffusion flux riving force Fick s first and second laws Interdiffusion Interstitial diffusion Self-diffusion Steady-state diffusion Nonsteady-state diffusion Temperature dependence of Vacancy diffusion 26

27 Homework #3: 5.7, 5.8, 5.22, 5.23, 5.2 ue date: Monday, September 20. Reading for next class: Chapter 6: Mechanical Properties of Metals Stress and Strain Tension Compression Shear Torsion Elastic deformation Plastic eformation Yield Strength Tensile Strength uctility Resilience Toughness Hardness Optional reading (not tested): details of the different types of hardness tests, variability of material properties (starting from the middle of page 74) 27

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